- Email: [email protected]
Vibration of fluid-conveying nanotubes subjected to magnetic field based on the thin-walled Timoshenko beam theory
Journal Pre-proof
Vibration of Fluid-Conveying Nanotubes Subjected to Magnetic Field Based on the Thin-Walled Timoshenko Beam Theory Mahta Ghane , Ali Reza Saidi , Reza Bahaadini PII: DOI: Reference:
S0307-904X(19)30716-4 https://doi.org/10.1016/j.apm.2019.11.034 APM 13164
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
23 May 2019 5 November 2019 19 November 2019
Please cite this article as: Mahta Ghane , Ali Reza Saidi , Reza Bahaadini , Vibration of FluidConveying Nanotubes Subjected to Magnetic Field Based on the Thin-Walled Timoshenko Beam Theory, Applied Mathematical Modelling (2019), doi: https://doi.org/10.1016/j.apm.2019.11.034
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Inc.
Highlights
Flutter instability of nanotubes conveying magnetic nanoflow is studied.
Nonlocal strain gradient thin-walled Timoshenko beam model are considered.
The effects of Knudsen number and magnetic nanoflow on the critical flutter velocity of nanotube are studied.
1
Vibration of Fluid-Conveying Nanotubes Subjected to Magnetic Field Based on the ThinWalled Timoshenko Beam Theory Mahta Ghane, Ali Reza Saidi, Reza Bahaadini* Department of Mechanical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran Abstract In this article, the flutter vibrations of fluid-conveying thin-walled nanotubes subjected to magnetic field is investigated. For modeling fluid structure interaction, the nonlocal strain gradient thin-walled Timoshenko beam model, Knudsen number and magnetic nanoflow are assumed. The Knudsen number is considered to analyse the slip boundary conditions between the fluid-flow and the nanotube’s wall, and the average velocity correction parameter is utilized to earn the modified flow velocity of nano-flow. Based on the extended Hamilton’s principle, the size-dependent governing equations and associated boundary conditions are derived. The coupled equations of motion are transformed to a general eigenvalue problem by applying extended Galerkin technique under the cantilever end conditions. The influences of nonlocal parameter, strain gradient length scale, magnetic nanoflow, longitudinal magnetic field, Knudsen number on the eigenvalues and critical flutter velocity of the nanotubes are studied. Keywords: Vibration; Nanotubes; Nonlocal Strain Gradient Theory; Magnetic Nanoflow; ThinWalled Beam.
*
Corresponding author. Tel.: +98-34-32111763, fax: +98-34-32120964 E-mail address: [email protected] (R. Bahaadini) 2
1. Introduction A remarkable point of view toward developing the different aspect of the carbon nanotubes (CNTs) characteristics in scientific fields was found after the CNTs were discovered by Iijima [1]. The CNTs, due to excellent mechanical properties, have many potential applications such as fluid storage, fluid transport, heat exchanger, drug delivery and hydrogen storage [2-5]. Vibration analysis of nanotubes conveying nanoflow has prepared strong and appropriate alternative fields to improve features of nanofluidic dynamic system research in nanobiological devices and nanomechanical systems such as fluid filtration devices, fluid transport, and targeted drug delivery devices. Based on these aspects, the nanotubes can be considered as biosensors for detection and elimination of cancer cells and become emerged as pharmaceutical excipients in the construction of versatile drug delivery systems. More specifically, they can be also used in order to deliver anticancer drugs into target site to kill metastatic cancer cells and have a great deal of characteristics for drug delivery applications to inject small-molecule drugs and some proteins [6]. Therefore, due to significant role of vibration behavior of nanotubes conveying nanoflow, a remarkable understanding of their vibration behavior can be necessary to exhibit better performance of them. Recently, nonlocal elasticity theory (NET), modified coupled stress theory (MCST), strain gradient theory(SGT), modified strain gradient theory (MSGT) and nonlocal strain gradient theory (NSGT) have been utilized to study the mechanical behaviors of the CNTs. In this regard, Ghavanloo and Fazelzadeh [7], based on NET, investigated the vibration behavior of CNTs conveying fluid. Bahaadini and Hosseini [8] studied static and dynamic instabilities of CNTs with flowing fluid under magnetic field. Rashidi et al. [9] performed the effect of small-sized on flow field of nanotube conveying nanoflow. Bahaadini and Hosseini [10] examined size-dependent eigenvalues and critical flutter velocity of CNTs containing fluid flow based on the NET. Challamel et al. [11] investigated the 3
self-adjointness of Eringen’s nonlocal elasticity based on simple one-dimensional beam model. Norouzzade and Ansari [12] performed a comparison study between the predictions of integral and differential nonlocal models for nanobeams under different kinds of end conditions. They also performed a finite element analysis of Timoshenko nano-beams based on the integral model of nonlocal continuum theory [13]. Ansari et al. [14] studied the free vibration and instability characteristics of nanoshells made of FGMs with internal fluid flow in thermal environment based on SGT. Yin et al. [15] investigated the vibration analysis of micro-pipes conveying fluid based on the MSGT. Hosseini and Bahaadini [16] inspected the stability behavior of sizedependent micro-pipes conveying fluid flow based on the MSGT. Ansari et al. [17] Studied the nonlinear vibration and instability of fluid-conveying SWBNNTs based on the MSGT. Based on the MCST, Dehrouyeh-Semnani et al. [18] performed the nonlinear dynamics of pipes containing fluid with non-uniform flow profile. Ansari et al. [19] studied the vibration and dynamic instability of cylindrical microshells made of functionally graded materials (FGMs) and containing flowing fluid based on MCST. Vibration and stability analysis of thin-walled fluidconveying pipe made of composite material was investigated by Bahaadini et al. [20]. Amabili and Garziera [21, 22] investigated the vibration of circular cylindrical shells with nonuniform edge constrains. Besides, there are many studies on the vibration and stability of micro/nano tubes conveying fluid [8, 23-31]. Recently, the nonlocal strain gradient theory which mixed the Aifantis’s strain gradient and Eringen’s nonlocal elasticity theories into a constitutive equation was introduced by Lim et al. [32]. They studied wave propagation in nano/micro beams model based on NSGT. Using NSGT, buckling [33], and vibration [34-36] of micro/nano structures have been studied. The linear and nonlinear vibrations of CNT in the context of NSGT were investigated in [37, 38].
4
Nonlinear vibration of nanobeam made of functionally graded materials by using NSGT was studied by Simsek [39]. Li and Hu [40] performed wave propagation in viscoelastic CNTs conveying fluid based on NSGT. By using nonlocal strain gradient Timoshenko and EulerBernoulli beams, Li et al. [41] inspected critical divergence velocity of micro-pipes conveying fluid. The influences of fluid mass ratio, surface effects, nonlocal and strain gradient parameters on the vibration of fluid- conveying nanotubes based on NSGT were examined by Atashafrooz et al. [42]. Choi et al. [43] performed the nonlinear vibration behavior of the thin-walled nano pipes conveying fluid. Vibration and instability of multi-walled CNTs conveying fluid were studied by Yun et al. [44] based on the NSGT. Dynamic stability and vibration of nanotubes conveying nanoflow based on NSGT were investigated by Bahaadini et al. [45, 46]. Besides, Bahaadini and Saidi [47] performed the stability of spinning thin-walled pipe reinforced with CNTs conveying fluid by using Galerkin method (GM). Dynamic instability analysis of the rotating thin-walled pipe reinforced with CNTs conveying fluid was analyzed by Bahaadini et al. [48]. Norouzzade et al. [49] studied the vibrational behavior of Timoshenko micro- and nano-beams By taking the nonlocal and strain gradient effects into account based on a novel size-dependent model. Norouzzade et al. [50] studied the static bending of nanoscale beams in the nonlinear regime based on NSGT. Norouzzade et al. [51] investigate the geometrically nonlinear wave propagation of nanobeams on the basis of the most comprehensive size dependent elasticity theory. Li et al. [52] investigated the size-dependent buckling analysis of nanobeams with a nonlocal strain gradient beam model incorporating the thickness effect. Tang et al. [53] investigate the static bending behavior of micro/nano-scale porous beams using a unified nonlocal strain gradient beam model with the thickness effect. Tang et al. [54] derived an
5
analytical solution for the bending frequencies of hinged-hinged nanobeams based on NSGT with the thickness effect. To the best of authors’ knowledge, no study investigated the vibration and flutter instability analysis of thin-walled nanotubes conveying magnetic nanoflow under magnetic field by using thin-walled Timoshenko beam theory. According to this theory, both flapping and lagging bending vibrations are considered and the proper components of the 3-D displacement vector are obtained. To formulate the size-dependent nanotubes conveying magnetic nanoflow, the nonlocal strain gradient thin-walled Timoshenko beam theory, magnetic nanoflow, Knudsen number and magnetic field are considered. The governing equations of motion are derived based on the extended Hamilton’s principle (EHP). Using the extended Galerkin method (EGM), the eigenvalue and flutter velocity are evaluated. The effects of strain gradient length scale, nonlocal parameter, Knudsen number and magnetic field on the natural frequencies and critical flutter velocity of the system are studied.
2. Theoretical formulation Based on NSGT, the total stress tensor is written as [32] ( )
where
(1)
denotes the one-dimensional differential operator, the classical stress tensor σ and the
higher-order stress tensor σ(1) are written as ( (
( (
) )
) )
(2)
( )
6
In equations (2) ε denote the classical strain tensor, ε represent the strain gradient tensor, C is the fourth-order elasticity tensor, constant and
is the material length scale parameter,
is an internal property of length. So,
is a material
displays the nonlocal parameter that
incorporates the small scale effects into the constitutive equations. At nanoscales, the material properties is size-dependent and the small scale effects of nanostructure have a crucial role in the dynamics of nanosystems. Therefore, it is necessary to use nonlocal theories incorporating small scale effects. Thus, the size-dependent constitutive equation for beams may be simplified as (
(
)
)
(3)
Fig. 1 is showed the model of the conveying-fluid thin-walled nanotube subjected to longitudinal magnetic field. the thin-walled nanotube has the length L, thickness h, outer diameter D, inner diameter d, and mass per unit length incompressible fluid of mass per unit length
. The nanotube conveying
, flowing axially with velocity
. In this study,
the nonlocal strain gradient thin-walled Timoshenko beam model is used to study vibration and stability analysis of the nanotube conveying nanoflow. Fig. 2 represents the kinematic variables and coordinate system of a thin-walled beam structure. The displacement components of thinwalled Timoshenko beams that undergo vertical bending with ignoring extension and torsion, can be written as follows [55, 56] (
)
(
),
(
)
(
),
(
)
(
)[ ( )
(4) ]
(
)[ ( )
7
]
In Eq. (4), the rigid body translations along the
- and
- axes are denoted by
(
), respectively. The rigid body rotations about the
(
) and
(
), respectively. The expressions for
- and (
) and
(
) and
- axes are represented by (
) can be written as
[56, 57] (
)
(
)
(
), (5)
(
)
(
where
and
)
(
)
are the shear strains and the partial derivative with respect to
is denoted by
superscript ( ). The non-zero strains components can be expressed as follows ,
,
,
The relations between strains of (
) and (
(6)
) in coordinate systems can be written as
,
(7)
,
Furthermore, the fluid particle velocity can be formulated as ([47] . ̇
/
. ̇
/
(
̇
̇
) ,
(8)
In Eq. (8), the subscript f shows the fluid and the superposed dot denotes the time derivative. The constitutive relations of thin-walled nanotubes based on NSGT can be formulated as
8
(
(
)
)
( [
)
]
(9) ][
[
]
where the reduced elastic coefficients can be written as ,
,
, (10)
, In Eq. (10), the Young and shear moduli and Poisson ratio of the nanotube are denoted by , ⁄
and , respectively. Furthermore,
denotes the shear correction coefficient.
The governing equations of motion and the corresponding boundary conditions are derived based on EHP which can be formulated as [58] ( ̇
∫ (
)
)
, (11)
at
,
In Eq. (11), the variation of the pipe kinetic energy, fluid kinetic energy and strain energy are denoted by
,
and
, respectively. Furthermore,
and
denote the position and
tangential vectors of a point on the free ended axis of the pipe, respectively. For thin-walled Timoshenko nanotubes, the virtual kinetic energy can be written as [55]
∫
∫
∫ {
̈
̈
[(
) ̈
(
) ̈ ]
[(
) ̈
(
) ̈ ]
9
(12) }
The virtual of kinetic energy of fluid flow is obtained as
∫
∫ { ,
∫
̈ ̇
̈
̈
,
̈ ̇
} (13)
∫ { , ̈
In Eq. (13),
and
̇ -
,
̈
̇ -
}
are the internal gyration radius about the x- and y- axes, respectively. The
virtual strain energy can be obtained as
)
∫ ∮ ∫ 0( ( )
.
( )
( )
( )
∫ ∮ ∫ 0.
( )
.
( )
where denoted by
⁄
/
/
/
1
/1
.
∫ ∮ 0.
( )
/
(14)
( )
( )
1
represents the one-dimensional differential operator. The classical stresses are ,
and
; the higher-order stresses are represented by
( )
,
( )
and
( )
.
By considering the components of strain and stress tensors obtained in Eqs. (9) and (14), the variation of strain is obtained as [55] 10
∫
{(
)
(
(
)
(
)
)
} (15)
( )
2. [
/
( )
. ( )
]
/
,
( )
( )
( )
3|
The components of Eq. (15) can be found in Appendix C. Considering the magnetic-fluid flow, the force applied to the nanotubes can be written as
( (
)
(16)
)
where the subscripts f and t correspond to the fluid and tube, respectively. In order to consider the small-size effect on fluid flow, a velocity correction factor ( (
by Rashidi et al. [9]. They used a modified nanotube by applying
)
(
where
)( (
is equal to 0.7 and
. In which
) was introduced as
/
(
)(
,
0.4 and
)(
)
)
as
(17)
is defined as: (
The magnetic force along the
)) (18)
. direction due to a longitudinal magnetic field based on the
Maxwell relation can be formulated as
11
4
where
5
(19)
represents the magnetic field permeability and
is the magnetic intensity of the
longitudinal magnetic field. By substituting Eqs. (12), (13), (15), (16) and (19) into Eq. (11), and using the arbitrary and independent properties of variations, the governing equations are derived as (
(
) (
)
) [(
) ̈
( ̇ (
δ
(
0
(
(
( )
(
(
)
(
) [(
) ̈
( ̇ (
(
)
(
)
)
) 1
( ]
)
(20)
(
) ̇ )
(
(
)
) (
)
) ̈
0
(
) ̇ )
( ( ̇
)
) 1
(
) ̇ )
(
)
(
) ]
(
)
(
)
)
)
(
(
) ̇ )
(
) ̈
( ̇
(
0(
(21) )
)
) (22)
(
)
(
) 1
(
) ̈
) [(
12
(
) ̈
̈ ]
) ̈
( (
) ̈
(
(
)
0(
̈
)
) (
(
)
) 1 (23)
(
) ̈
) [( ) ̈
(
) ̈
( ) ̈
(
̈ ] ̈
Besides, the boundary conditions are: at z=0:
(24)
at
( ) ( )
( )
( )
( )
( )
(25)
In equations of motion, the stiffness and reduced mass terms are represented by
and
,
respectively, which are defined in Appendix A.
3. Solution method The EGM is utilized to convert the partial differential equations into ordinary differential equations. Therefore, the following expansions are assumed: 13
̅ (
)
̅ (
,
)
, (26)
̅ (
)
̅ (
,
)
,
In Eqs. (26), the shape functions which satisfy the boundary conditions are represented by and
. The generalized coordinates vectors are denoted by
,
,
and
, , . The
following equation can be obtained by using EGM , - ̈( )
, - ̇( )
, - ( )
(27)
,
In Eq. (27), mass, damping and stiffness matrices are denoted as , - ,, - and , -, respectively. These matrices are defined in Appendix B. Defining the state vector as
( )
* ( ) ̇ ( )+ , Eq. (27) can be converted to the first-order
state space form as: ̇( )
, - ( ),
(28)
where , -
[
, , - , -
, ], , - , -
Substituting the state vector
( )
(29) ̅
(
)
into Eq. (28), the following equation can be
expressed: (, -
, -) ̅
The eigenvalue
(30)
. ( )
is a complex quantity, i.e.,
( ). The types of instability are
determined based on the sign of real and imaginary parts of the complex eigenvalue [27, 59].
14
4. Results and discussion The natural frequency and flutter velocity of conveying-fluid nanotubes subjected to longitudinal magnetic field based on the nonlocal strain gradient thin-walled Timoshenko theory are studied. The effects of fluid velocity, Knudsen number, strain gradient length scale, nonlocal parameter and magnetic field on the natural frequency and flutter velocity of the system are investigated. The dimensionless frequency as a function of fluid velocity for cantilever thinwalled nanotube conveying fluid is illustrated in Fig. 2 and the results are compared with those reported by Yun et al. [44] based on the classical thin-walled Timoshenko CNT. Fig. 2, shows that the present results are in excellent agreement with the reported data. In order to illustrate the accuracy of present study, the natural frequency of nanotube conveying fluid is compared those obtained using classical continuum theory. Also, for validation purposes and to demonstrate the accuracy of suggested model, the eigenfrequencies obtained from the current study are compared with those reported by Apuzzo et al. [60] in Table 4. Furthermore, the convergence of eigenvalues of thin-walled nanotube conveying fluid is shown in Table 1 for 𝛽 μ results at
, 𝜉
and and
,
,
. It should be noted that the small difference between the show that the eigenvalue with
is suitably accurate to study
the vibration and stability analysis of cantilever nanotube conveying nanflow. The effects of and Kn on the critical flutter velocity for different classical and non-classical theories is observed in Tables 2 and 3, respectively. It can be found that the critical flutter velocity increases by increasing the magnetic intensity
. While by increasing the Knudsen number, the critical
flutter velocity is decreased. Moreover, the maximum and minimum flutter velocities are predicted by using the SGT and NT Timoshenko thin-walled pipe models, respectively.
15
It should be noted that the non-classical continuum theories predict a more precise mechanical model on the nano-scale materials. Hence, the effects of non-classical continuum theories on the eigenvalues and critical flutter velocity are illustrated in Figs. 3 through 5. Furthermore, NSGT can be converted to a strain gradient theory (SGT) if 𝜉 nonlocal theory (NT) if 𝜉
and
and a classical theory (CT) if 𝜉
and and
,a .
Besides, it is seen that the natural frequencies and critical flutter velocity of the conveying-fluid thin-walled nanotubes based on the NSGT can be smaller than that of classical model, which can be described by the stiffness-softening effect owing to the values of scaling parameters. The stiffness-softening effects are seen when 𝜉
. In Fig. 3, the natural frequencies and flutter
velocity decrease by increasing the nonlocal parameter which means that considering the nonlocal effect leads to decrease the bending rigidity of the nanotubes. Besides in Fig. 4, the natural frequencies and flutter velocity increase by increasing the strain gradient parameter. Fig. 5 shows the effects of both strain gradient length scale and nonlocal parameters on the natural frequency and flutter velocity of the nanotubes containing magnetic nanoflow for 𝛽 and
,
. The results show that the natural frequency and flutter velocity decrease
as the nonlocal parameter increases which means that considering the nonlocal effect leads to decreasing the bending rigidity of the carbon nanotubes. Furthermore, the natural frequency and flutter velocity increases by increasing the strain gradient length scale. It means that the stiffness of the system increases, as the strain gradient length scale increases. Furthermore, the strain gradient and the nonlocal elasticity models predict stiffness enhancement effect and stiffness softening effect for the nanotube, respectively. The results show that the vibration and stability analyses of the thin-walled nanotubes based on the nonlocal strain gradient model may be more reliable.
16
The effect of aspect ratio ( ⁄ ) on the eigenvalues of thin-walled nanotubes conveying magnetic nanoflow under longitudinal magnetic field is investigated in Fig. 6. This figure is presented for 𝛽
𝜉
and
. The results show that the aspect
ratio effect tends to increase the bending stiffness of the thin-walled nanotubes and the natural frequencies and flutter velocity increase. The effects of intensity of longitudinal magnetic field on the eigenvalues of thin-walled nanotubes conveying magnetic nanoflow are studied in Fig. 7. The results are presented for 𝛽
,
and
𝜉
. The results indicated that the
natural frequencies and flutter velocity increase by increasing the intensity of longitudinal magnetic field. In other words, the results show that the magnetic fluid tends to increase the bending stiffness of the system and the natural frequency and flutter velocity increase. Figs. 8 and 9 shows the influence of Knudsen number on the natural frequencies and flutter velocity of magnetic nanoflow conveying-nanotube based on the NSGT for and 𝜉
. It is noted that the Helium gas is considered in Fig. 9. The Knudsen
number is used to identify different flow regimes such as the continuum flow regime ( ), the slip flow regime ( and the free molecular flow regime (
), the transition flow regime (
),
). For the continuum flow regime, it is revealed that
the natural frequencies and flutter velocity are not changed as the Knudsen number increases. Also, the natural frequencies and flutter velocity decrease, as the Knudsen number increases in the slip flow regime. Besides, the natural frequencies and flutter velocity decrease as the Knudsen number increases in the transient flow regime. For
, the natural frequencies
and flutter velocity tends to vanish and the effect of flow velocity on the vibration and stability analysis becomes faint. Fig. 10 shows the effects of mass fluid ratio on the eigenvalues of the
17
conveying-fluid nanotubes for
,
and
𝜉
. The results illustrate that the
natural frequency and flutter velocity increase as the mass fluid ratio increases. Fig. 11 shows the critical flutter velocity in terms of size-dependent parameters for different non-classical thin-walled beam models. In this figure, the values
.001, 𝛽
are considered. The results can be shown that the thin-walled pipe with SGT and NT models have the highest and lowest critical flutter velocity among others. Also, the results for thin-walled pipe with NSGT are tend to get closer to those of SGT or NT models due to size-dependent parameters. In this Figure, the flutter boundaries separate the stable and unstable areas. It should be noted that, in unstable area any initial dynamic structural disturbance will grow up till the mechanical failure will happen. Fig. 12 shows the effects of both strain gradient and nonlocal parameters on the natural frequency of the CNTs conveying nanoflow for
,𝛽
. It is clear that
the natural frequency decreases by increasing nonlocal parameter which means that the considering nonlocal elasticity theory leads to decrease the bending rigidity of the CNTs. Whereas, by increasing the strain gradient parameter, the stiffness of the CNTs increases, and consequently, the natural frequency increases.
5. Conclusion The size-dependent stability analysis of the conveying-fluid thin-walled nanotubes under longitudinal magnetic field was investigated. Using NSGT, thin-walled Timoshenko beam theory, Knudsen number and magnetic nanoflow, the equations of motion were derived based on the EHP. Using the EGM, these equations were discretized. The influences of strain gradient 18
length scale, nonlocal parameter, fluid velocity, Knudsen number and magnetic field on the vibration and stability analysis of the system were discussed. It was observed that the magnetic fluid has considerable effect on the natural frequency and flutter velocity of the nanotubes. Specifically, this study showed that the magnetic field effects increase the critical flow velocity, thus makes the system more stable. It was demonstrated that as the Knudsen number rises, the natural frequency and critical flutter velocity decreases. Besides, it was shown that the natural frequency and critical flow velocity decrease with increasing nonlocal parameter, whereas those have the opposite effect with increasing strain gradient parameter. Furthermore, it was seen that in the NSGT, if the nonlocal coefficient be greater than the strain gradient parameter, by increasing the nonlocal strain gradient parameter, the natural frequency and consequently the critical flow velocity decreases; otherwise, it increases. It was illustrated that increasing the fluid mass ratio and aspect ratio of the CNT increase the natural frequency of the system, which makes the system more stable.
Appendix A The global stiffness quantities
counting the pretwist effect can be expressed as bellows
∮
∮(
∮4
)
(
∮(
) 5
)
∮4
19
(
)
(A.1)
(
) 5
∮4
(
) 5
∮4
(
)
(
) 5
where
(A.2) {
}
*
∫
+
Furthermore, the mass terms (
)
∮
(
(
)
∮
4.
(
)
∫
(
are presented as )
/ . / . / 5
(A.3)
,
)
Appendix B To simplify the analysis, we introduce the following dimensionless parameters as below ̅
̅ ̅
̅ ̅
̅
𝛽
√
(
(B.1)
√
)
𝛽
(
20
)
(
)
√
(
(
)
)
√ 𝜉
The matrices are defined as follows
, -
, [
]
, [
] (B.2)
, [
(𝛽
𝛽)
∫ * + * +
]
(
) (𝛽
𝛽)
∫ * + * +
(B.3) √𝛽 (
)
∫ * + * +
(
) 𝛽
∫ * +
* +
∫ * + * +
21
∫ * + * +
∫ (* + * + (
) [
* +
∫ .{
(𝛽
√𝛽 (
* + * +
* +
(
} * +
{
∫ * + * +
* + * +
* + * + ]
* + * + )
} * + /
(
(
) 𝛽
) (𝛽
𝛽)
∫ * +
* +
∫ * + * +
∫ * + * +
∫ * + * +
)
∫ (* + * + (
) * + * +
) * + * +
∫ * + * +
𝛽)
(
) [
* +
∫ (*
(
* + * +
* +
+ * +
) ∫ .{
(
*
(
) * + * +
) * + * +
* + * +
* + * + ]
* + * + )
+ * + )
} {
}
(
)
(
22
){
} {
}/
+ {
∫ .*
∫ .
{
} {
}
∫ .* + {
) ∫ (*
+ *
}
* +
} {
}/
0{
{
} {
}
{
}
{
} 1/
+
*
+ 1/
}/
+
}
{
+ *
} *
∫ .{
*
} {
+ {
}
{
(
∫ .
{
) *
(
}
∫ * + {
∫ (
}
+
*
+
(
(
+ *
) {
+
)
}
{
(
} )
)*
} *
+/
0*
+ *
23
+
+ *
+)
*
∫ (* + *
+
* +
∫ * + *
*
∫ (
*
+)
+
+ *
+
*
+
*
+ )
Appendix C The stress resultants are defined as follows
(
)
(
∫
)
.
( )
( )
/
∫
( )
(
) (C.1)
(
)
Where
(
∫
)
( )
.
( )
/
.
∫
( )
denotes the axial normal stress and ( )
( )
/
denotes the shear stress
,
( )
(C.2)
( )
Moreover, the other stress resultants and the stress-couples can be defined as (
)
( )
∮
(
)
∮
( )
(C.3) (
)
∮0
( )
( )
1
( )
(
)
24
∮0
( )
( )
( )
( )
1
(
)
∮0
(
)
∮6
(
)
∮6
where
( )
( )
( )
( )
1
7
( )
7
)
∮0
( )
( )
( )
( )
( )
(
)
∮6
( )
( )
( )
( )
(
)
∮6
( )
( )
( )
corresponds to the axial force;
and y- directions, respectively;
(
and
and
1
7 7
denote associated with the shear forces in xrepresent the bending moments of x- and y-
directions, respectively.
25
References [1] S. Iijima, Helical microtubules of graphitic carbon, nature, 354 (1991) 56. [2] G. Hummer, J.C. Rasaiah, J.P. Noworyta, Water conduction through the hydrophobic channel of a carbon nanotube, Nature, 414 (2001) 188. [3] Y. Gao, Y. Bando, Nanotechnology: carbon nanothermometer containing gallium, Nature, 415 (2002) 599. [4] R.F. Gibson, E.O. Ayorinde, Y.F. Wen, Vibrations of carbon nanotubes and their composites: A review, Composites Science and Technology, 67 (2007) 1-28. [5] D. Mattia, Y. Gogotsi, Review: static and dynamic behavior of liquids inside carbon nanotubes, Microfluidics and Nanofluidics, 5 (2008) 289-305. [6] D. Pantarotto, C.D. Partidos, J. Hoebeke, F. Brown, E. Kramer, J.P. Briand, S. Muller, M. Prato, A. Bianco, Immunization with peptide-functionalized carbon nanotubes enhances virusspecific neutralizing antibody responses, Chemistry & biology, 10 (2003) 961-966. [7] E. Ghavanloo, S.A. Fazelzadeh, Flow-thermoelastic vibration and instability analysis of viscoelastic carbon nanotubes embedded in viscous fluid, Physica E: Low-dimensional Systems and Nanostructures, 44 (2011) 17-24. [8] R. Bahaadini, M. Hosseini, Nonlocal divergence and flutter instability analysis of embedded fluid-conveying carbon nanotube under magnetic field, Microfluidics and Nanofluidics, 20 (2016) 108. [9] V. Rashidi, H.R. Mirdamadi, E. Shirani, A novel model for vibrations of nanotubes conveying nanoflow, Computational Materials Science, 51 (2012) 347-352. [10] R. Bahaadini, M. Hosseini, Effects of nonlocal elasticity and slip condition on vibration and stability analysis of viscoelastic cantilever carbon nanotubes conveying fluid, Computational Materials Science, 114 (2016) 151-159. [11] N. Challamel, Z. Zhang, C.M. Wang, J.N. Reddy, Q. Wang, T. Michelitsch, B. Collet, On nonconservativeness of Eringen’s nonlocal elasticity in beam mechanics: correction from a discrete-based approach, Archive of Applied Mechanics, 84 (2014) 1275-1292. [12] A. Norouzzadeh, R. Ansari, H. Rouhi, Pre-buckling responses of Timoshenko nanobeams based on the integral and differential models of nonlocal elasticity: An isogeometric approach, Applied Physics A, 123 (2017) 330. [13] A. Norouzzadeh, R. Ansari, Finite element analysis of nano-scale Timoshenko beams using the integral model of nonlocal elasticity, Physica E: Low-dimensional Systems and Nanostructures, 88 (2017) 194-200. [14] R. Ansari, R. Gholami, A. Norouzzadeh, Size-dependent thermo-mechanical vibration and instability of conveying fluid functionally graded nanoshells based on Mindlin's strain gradient theory, Thin-Walled Structures, 105 (2016) 172-184. [15] L. Yin, Q. Qian, L. Wang, Strain gradient beam model for dynamics of microscale pipes conveying fluid, Applied Mathematical Modelling, 35 (2011) 2864-2873. [16] M. Hosseini, R. Bahaadini, Size dependent stability analysis of cantilever micro-pipes conveying fluid based on modified strain gradient theory, International Journal of Engineering Science, 101 (2016) 1-13. [17] R. Ansari, A. Norouzzadeh, R. Gholami, M.F. Shojaei, M. Hosseinzadeh, Size-dependent nonlinear vibration and instability of embedded fluid-conveying SWBNNTs in thermal environment, Physica E: Low-dimensional Systems and Nanostructures, 61 (2014) 148-157.
26
[18] A.M. Dehrouyeh-Semnani, H. Zafari-Koloukhi, E. Dehdashti, M. Nikkhah-Bahrami, A parametric study on nonlinear flow-induced dynamics of a fluid-conveying cantilevered pipe in post-flutter region from macro to micro scale, International Journal of Non-Linear Mechanics, 85 (2016) 207-225. [19] R. Ansari, R. Gholami, A. Norouzzadeh, S. Sahmani, Size-dependent vibration and instability of fluid-conveying functionally graded microshells based on the modified couple stress theory, Microfluidics and nanofluidics, 19 (2015) 509-522. [20] R. Bahaadini, M.R. Dashtbayazi, M. Hosseini, Z. Khalili-Parizi, Stability analysis of composite thin-walled pipes conveying fluid, Ocean Engineering, 160 (2018) 311-323. [21] M. Amabili, R. Garziera, Vibrations of circular cylindrical shells with nonuniform constraints, elastic bed and added mass; Part I: empty and fluid-filled shells, Journal of Fluids and Structures, 14 (2000) 669-690. [22] M. Amabili, R. Garziera, Vibrations of circular cylindrical shells with nonuniform constraints, elastic bed and added mass. Part II: shells containing or immersed in axial flow, Journal of Fluids and Structures, 16 (2002) 31-51. [23] M. Hosseini, R. Bahaadini, M. Makkiabadi, Application of the Green function method to flow-thermoelastic forced vibration analysis of viscoelastic carbon nanotubes, Microfluidics and Nanofluidics, 22 (2018) 6. [24] R. Bahaadini, M. Hosseini, Flow-induced and mechanical stability of cantilever carbon nanotubes subjected to an axial compressive load, Applied Mathematical Modelling, 59 (2018) 597-613. [25] R. Bahaadini, A.R. Saidi, M. Hosseini, Dynamic stability of fluid-conveying thin-walled rotating pipes reinforced with functionally graded carbon nanotubes, Acta Mechanica, 229 (2018) 5013-5029. [26] M. Hosseini, A.Z.B. Maryam, R. Bahaadini, Forced vibrations of fluid-conveyed double piezoelectric functionally graded micropipes subjected to moving load, Microfluidics and Nanofluidics, 21 (2017) 134. [27] R. Bahaadini, M. Hosseini, A. Jamalpoor, Nonlocal and surface effects on the flutter instability of cantilevered nanotubes conveying fluid subjected to follower forces, Physica B: Condensed Matter, 509 (2017) 55-61. [28] M. Mohammadimehr, M. Mehrabi, Electro-thermo-mechanical vibration and stability analyses of double-bonded micro composite sandwich piezoelectric tubes conveying fluid flow, Applied Mathematical Modelling, 60 (2018) 255-272. [29] B. Wang, Z. Deng, H. Ouyang, X. Xu, Free vibration of wavy single-walled fluid-conveying carbon nanotubes in multi-physics fields, Applied Mathematical Modelling, 39 (2015) 67806792. [30] M. Hosseini, M. Sadeghi-Goughari, Vibration and instability analysis of nanotubes conveying fluid subjected to a longitudinal magnetic field, Applied Mathematical Modelling, 40 (2016) 2560-2576. [31] A. Ghorbanpour Arani, R. Kolahchi, Z. Khoddami Maraghi, Nonlinear vibration and instability of embedded double-walled boron nitride nanotubes based on nonlocal cylindrical shell theory, Applied Mathematical Modelling, 37 (2013) 7685-7707. [32] C.W. Lim, G. Zhang, J.N. Reddy, A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation, Journal of the Mechanics and Physics of Solids, 78 (2015) 298-313.
27
[33] A. Farajpour, M.H. Yazdi, A. Rastgoo, M. Mohammadi, A higher-order nonlocal strain gradient plate model for buckling of orthotropic nanoplates in thermal environment, Acta Mechanica, 227 (2016) 1849-1867. [34] Y. Tang, Y. Liu, D. Zhao, Viscoelastic wave propagation in the viscoelastic single walled carbon nanotubes based on nonlocal strain gradient theory, Physica E: Low-dimensional Systems and Nanostructures, 84 (2016) 202-208. [35] H. Zeighampour, Y. Tadi Beni, I. Karimipour, Material length scale and nonlocal effects on the wave propagation of composite laminated cylindrical micro/nanoshells, The European Physical Journal Plus, 132 (2017) 503. [36] L. Lu, X. Guo, J. Zhao, Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory, International Journal of Engineering Science, 116 (2017) 12-24. [37] X. Li, Y. Li, Y. Qin, Free vibration characteristics of a spinning composite thin-walled beam under hygrothermal environment, International Journal of Mechanical Sciences, 119 (2016) 253-265. [38] R. Fernandes, S. El-Borgi, S. Mousavi, J.N. Reddy, A. Mechmoum, Nonlinear sizedependent longitudinal vibration of carbon nanotubes embedded in an elastic medium, Physica E: Low-dimensional Systems and Nanostructures, 88 (2017) 18-25. [39] M. Şimşek, Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach, International Journal of Engineering Science, 105 (2016) 12-27. [40] L. Li, Y. Hu, Wave propagation in fluid-conveying viscoelastic carbon nanotubes based on nonlocal strain gradient theory, Computational Materials Science, 112 (2016) 282-288. [41] M. Mohammadimehr, M.J. Farahi, S. Alimirzaei, Vibration and wave propagation analysis of twisted micro-beam using strain gradient theory, Applied Mathematics and Mechanics, 37 (2016) 1375-1392. [42] M. Atashafrooz, R. Bahaadini, H.R. Sheibani, Nonlocal, strain gradient and surface effects on vibration and instability of nanotubes conveying nanoflow, Mechanics of Advanced Materials and Structures, (2018) 1-13. [43] J. Choi, O. Song, S.K. Kim, Nonlinear stability characteristics of carbon nanotubes conveying fluids, Acta Mechanica, 224 (2013) 1383-1396. [44] K. Yun, J. Choi, S.K. Kim, O. Song, Flow-induced vibration and stability analysis of multiwall carbon nanotubes, Journal of mechanical science and technology, 26 (2012) 3911-3920. [45] R. Bahaadini, A.R. Saidi, M. Hosseini, On dynamics of nanotubes conveying nanoflow, International Journal of Engineering Science, 123 (2018) 181-196. [46] R. Bahaadini, A.R. Saidi, M. Hosseini, Flow-induced vibration and stability analysis of carbon nanotubes based on the nonlocal strain gradient Timoshenko beam theory, Journal of Vibration and Control, 21 (2018) 203-218. [47] R. Bahaadini, A.R. Saidi, Stability analysis of thin-walled spinning reinforced pipes conveying fluid in thermal environment, European Journal of Mechanics-A/Solids, 72 (2018) 298-309. [48] R. Bahaadini, A.R. Saidi, M. Hosseini, Dynamic stability of fluid-conveying thin-walled rotating pipes reinforced with functionally graded carbon nanotubes, Acta Mechanica, (2018) 117. [49] A. Norouzzadeh, R. Ansari, H. Rouhi, Isogeometric vibration analysis of small-scale Timoshenko beams based on the most comprehensive size-dependent theory, Scientia Iranica, 25 (2018) 1864-1878. 28
[50] A. Norouzzadeh, R. Ansari, H. Rouhi, Nonlinear bending analysis of nanobeams based on the nonlocal strain gradient model using an isogeometric finite element approach, Iranian Journal of Science and Technology, Transactions of Civil Engineering, 43 (2019) 533-547. [51] A. Norouzzadeh, R. Ansari, H. Rouhi, Nonlinear wave propagation analysis in Timoshenko nano-beams considering nonlocal and strain gradient effects, Meccanica, 53 (2018) 3415-3435. [52] L. Li, H. Tang, Y. Hu, The effect of thickness on the mechanics of nanobeams, International Journal of Engineering Science, 123 (2018) 81-91. [53] H. Tang, L. Li, Y. Hu, Coupling effect of thickness and shear deformation on sizedependent bending of micro/nano-scale porous beams, Applied Mathematical Modelling, 66 (2019) 527-547. [54] H. Tang, L. Li, Y. Hu, W. Meng, K. Duan, Vibration of nonlocal strain gradient beams incorporating Poisson's ratio and thickness effects, Thin-Walled Structures, 137 (2019) 377-391. [55] L. Librescu, O. Song, Thin-walled composite beams: theory and application, Springer Science & Business Media 2005. [56] R. Bahaadini, A.R. Saidi, On the stability of spinning thin-walled porous beams, ThinWalled Structures, 132 (2018) 604-615. [57] L. Librescu, S.Y. Oh, O. Song, Thin-walled beams made of functionally graded materials and operating in a high temperature environment: vibration and stability, Journal of Thermal Stresses, 28 (2005) 649-712. [58] M.P. Paidoussis, Fluid-structure interactions: slender structures and axial flow, Academic press1998. [59] R. Bahaadini, A.R. Saidi, Aeroelastic analysis of functionally graded rotating blades reinforced with graphene nanoplatelets in supersonic flow, Aerospace Science and Technology, 80 (2018) 381-391. [60] A. Apuzzo, R. Barretta, S. Faghidian, R. Luciano, F.M. de Sciarra, Free vibrations of elastic beams by modified nonlocal strain gradient theory, International Journal of Engineering Science, 133 (2018) 99-108.
29
Table Captions Table 1. The convergence of eigenvalues of nanotubes conveying nanflow (for β ,𝜉
,μ
and
,
).
Table 2. The critical flutter velocity of cantilever nanotubes conveying magnetic nanoflow (for 𝛽 , , 𝜉 ). Table 3. The critical flutter velocity of cantilever nanotubes conveying magnetic nanoflow (for 𝛽 , , 𝜉 ). Table 4. The comparison of the frequencies obtained from the present model with the results presented by Apuzzo et al. [60].
Figure Captions Fig. 1. Schematic of a nanotube conveying magnetic nanoflow under magnetic field. Fig. 2. The comparison between the results of present study and those reported by Yun et al. [44]. Fig. 3. The variation of (a) natural frequency (b) and real part of eigenvalues for 𝛽 𝜉
,
.
Fig. 4. The variation of (a) natural frequency (b) and real part of eigenvalues for 𝛽
Fig. 5. The Argand diagram of the thin-walled carbon nanotubes for 𝛽
,
and
. Fig. 6. The variation of (a) natural frequency (b) and real part of eigenvalues for 𝛽 ,𝜉
,
and
.
Fig. 7. The variation of (a) natural frequency (b) and real part of eigenvalues for 𝛽 and
𝜉
,
.
Fig. 8. The variation of (a) natural frequency (b) and real part of eigenvalues for 𝛽 . 30
,
Fig. 9. The variation of (a) natural frequency (b) and real part of eigenvalues for and 𝜉
,
.
Fig. 10. The Argand diagram of the thin-walled carbon nanotubes for 𝜉
,
,
and
.
Fig. 11. The critical flutter velocity in terms of size-dependent parameters for thin-walled carbon nanotubes ( .001, 𝛽 ). Fig. 12. Variation of natural frequency versus nanlocal and strain gradient parameters for thinwalled carbon nanotubes ( ,𝛽 ).
Table 1. The convergence of eigenvalues of nanotubes conveying nanflow (for 𝛽 ,𝜉
,
and
).
N 4
3.392695-i2.096368
20.607984-i1.850850
58.835530-i1.916890
5
3.384872-i2.099706
20.521909-i1.882895
58.381143-i1.792144
6
3.387659-i2.098219
20.485642-i1.871531
57.945738-i1.865083
7
3.386026-i2.099021
20.480879-i1.875274
57.896421-i1.849693
8
3.386670-i2.098572
20.472613-i1.873468
57.833426-i1.863386
9
3.386631-i2.098680
20.470710-i1.873991
57.821514-i1.863896
10
3.386359-i2.098791
20.471034-i1.873763
57.806043-i1.862834
31
,
Table 2. The critical flutter velocity of cantilever nanotubes conveying magnetic nanoflow (for 𝛽 , , 𝜉 ). ( ) CT
NT
SGT
NSGT
0
4.3346
4.3168
4.3627
4.4239
50
4.6824
4.6657
4.7084
4.7650
100
5.5972
5.5829
5.6119
5.6662
150
6.8553
6.8432
6.8730
6.9111
Table 3. The critical flutter velocity of cantilever nanotubes conveying magnetic nanoflow (for 𝛽 , , 𝜉 ).
CT
NT
SGT
NSGT
0
4.3346
4.3168
4.3627
4.4239
0.001
4.3042
4.2865
4.3321
4.3928
0.01
4.0708
4.0540
4.0972
4.1546
0.1
3.0280
3.0156
3.0476
3.0903
Table 4. The comparison of the frequencies obtained from the present model with the results presented by Apuzzo et al. [60]. Natural frequencies Present study μ 0 0. 2 0. 4 0. 6 0. 8 1
𝜉 1.0084595 3 0.7352661 6 0.5723993 4 0.4673513 5 0.3939144 4 0.3400715 8
𝜉 1.096907 0.834476 2 0.670989 8 0.560478 1 0.480777 0.420533
Apuzzo et al. [60]
𝜉
𝜉
1.164200 6 0.910181 8 0.751318 2 0.641014 3 0.559740 3 0.496819 3
1.201338 8 0.954117 9 0.800682 4 0.693235 1 0.613287 1 0.550564 3
𝜉 1.227948 2 0.987460 9 0.839662 2 0.736334 4 0.659219 7 0.598238 4
32
𝜉
𝜉
𝜉
𝜉
𝜉
1.01778
1.10709
1.17522
1.21277
1.24021
0.74257 8
0.84290 5 0.67822 2 0.56680 7 0.48660 5 0.42618 1
0.91996 6 0.75956 3 0.64858 3 0.56674 8 0.50369 8
0.96459 1 0.80955 9 0.70163 7 0.62122 3 0.55853 1
0.99841 9 0.84912 8 0.74540 2 0.66789 4 0.60714 2
0.57857 0.47234 1 0.39849 3 0.34436 4
Fig. 1. Schematic of a nanotube conveying magnetic nanoflow under magnetic field.
Fig. 2. The comparison between the results of present study and those reported by Yun et al.
33
[44].
(a)
(b) Fig. 3. The variation of (a) natural frequency (b) and real part of eigenvalues for 𝛽 34
,
𝜉
.
(a)
(b) Fig. 4. The variation of (a) natural frequency (b) and real part of eigenvalues for 𝛽 35
,
.
Fig. 5. The Argand diagram of the thin-walled carbon nanotubes for 𝛽 .
36
and
(a)
(b) Fig. 6. The variation of (a) natural frequency (b) and real part of eigenvalues for 𝛽 𝜉
and
.
37
(a)
(b) Fig. 7. The variation of (a) natural frequency (b) and real part of eigenvalues for 𝛽 𝜉
38
,
(a)
(b) Fig. 8. The variation of (a) natural frequency (b) and real part of eigenvalues for 𝛽 .
39
(a)
(b) Fig. 9. The variation of (a) natural frequency (b) and real part of eigenvalues for .
40
,
Fig. 10. The Argand diagram of the thin-walled carbon nanotubes for 𝜉
.
41
and
7 = 0, (NT) = 0, (SGT) = 0.1, (NSGT) = 0.1, (NSGT)
6.8 6.6
Flutter
Critical flutter velocity, (ucr )
6.4 6.2 6
5.8 5.6 5.4 5.2 5
Stable
4.8 4.6 0
0.05
0.1
0.15
0.2
Size-dependent parameters, (,)
Fig. 11. The critical flutter velocity in terms of size-dependent parameters for thin-walled carbon nanotubes ( ,𝛽 ).
42
Fig. 12. Variation of natural frequency versus nanlocal and strain gradient parameters for thinwalled carbon nanotubes ( ,𝛽 ).
43
Vibration of Fluid-Conveying Nanotubes Subjected to Magnetic Field Based on the Thin-Walled Timoshenko Beam Theory Mahta Ghane , Ali Reza Saidi , Reza Bahaadini PII: DOI: Reference:
S0307-904X(19)30716-4 https://doi.org/10.1016/j.apm.2019.11.034 APM 13164
To appear in:
Applied Mathematical Modelling
Received date: Revised date: Accepted date:
23 May 2019 5 November 2019 19 November 2019
Please cite this article as: Mahta Ghane , Ali Reza Saidi , Reza Bahaadini , Vibration of FluidConveying Nanotubes Subjected to Magnetic Field Based on the Thin-Walled Timoshenko Beam Theory, Applied Mathematical Modelling (2019), doi: https://doi.org/10.1016/j.apm.2019.11.034
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Inc.
Highlights
Flutter instability of nanotubes conveying magnetic nanoflow is studied.
Nonlocal strain gradient thin-walled Timoshenko beam model are considered.
The effects of Knudsen number and magnetic nanoflow on the critical flutter velocity of nanotube are studied.
1
Vibration of Fluid-Conveying Nanotubes Subjected to Magnetic Field Based on the ThinWalled Timoshenko Beam Theory Mahta Ghane, Ali Reza Saidi, Reza Bahaadini* Department of Mechanical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran Abstract In this article, the flutter vibrations of fluid-conveying thin-walled nanotubes subjected to magnetic field is investigated. For modeling fluid structure interaction, the nonlocal strain gradient thin-walled Timoshenko beam model, Knudsen number and magnetic nanoflow are assumed. The Knudsen number is considered to analyse the slip boundary conditions between the fluid-flow and the nanotube’s wall, and the average velocity correction parameter is utilized to earn the modified flow velocity of nano-flow. Based on the extended Hamilton’s principle, the size-dependent governing equations and associated boundary conditions are derived. The coupled equations of motion are transformed to a general eigenvalue problem by applying extended Galerkin technique under the cantilever end conditions. The influences of nonlocal parameter, strain gradient length scale, magnetic nanoflow, longitudinal magnetic field, Knudsen number on the eigenvalues and critical flutter velocity of the nanotubes are studied. Keywords: Vibration; Nanotubes; Nonlocal Strain Gradient Theory; Magnetic Nanoflow; ThinWalled Beam.
*
Corresponding author. Tel.: +98-34-32111763, fax: +98-34-32120964 E-mail address: [email protected] (R. Bahaadini) 2
1. Introduction A remarkable point of view toward developing the different aspect of the carbon nanotubes (CNTs) characteristics in scientific fields was found after the CNTs were discovered by Iijima [1]. The CNTs, due to excellent mechanical properties, have many potential applications such as fluid storage, fluid transport, heat exchanger, drug delivery and hydrogen storage [2-5]. Vibration analysis of nanotubes conveying nanoflow has prepared strong and appropriate alternative fields to improve features of nanofluidic dynamic system research in nanobiological devices and nanomechanical systems such as fluid filtration devices, fluid transport, and targeted drug delivery devices. Based on these aspects, the nanotubes can be considered as biosensors for detection and elimination of cancer cells and become emerged as pharmaceutical excipients in the construction of versatile drug delivery systems. More specifically, they can be also used in order to deliver anticancer drugs into target site to kill metastatic cancer cells and have a great deal of characteristics for drug delivery applications to inject small-molecule drugs and some proteins [6]. Therefore, due to significant role of vibration behavior of nanotubes conveying nanoflow, a remarkable understanding of their vibration behavior can be necessary to exhibit better performance of them. Recently, nonlocal elasticity theory (NET), modified coupled stress theory (MCST), strain gradient theory(SGT), modified strain gradient theory (MSGT) and nonlocal strain gradient theory (NSGT) have been utilized to study the mechanical behaviors of the CNTs. In this regard, Ghavanloo and Fazelzadeh [7], based on NET, investigated the vibration behavior of CNTs conveying fluid. Bahaadini and Hosseini [8] studied static and dynamic instabilities of CNTs with flowing fluid under magnetic field. Rashidi et al. [9] performed the effect of small-sized on flow field of nanotube conveying nanoflow. Bahaadini and Hosseini [10] examined size-dependent eigenvalues and critical flutter velocity of CNTs containing fluid flow based on the NET. Challamel et al. [11] investigated the 3
self-adjointness of Eringen’s nonlocal elasticity based on simple one-dimensional beam model. Norouzzade and Ansari [12] performed a comparison study between the predictions of integral and differential nonlocal models for nanobeams under different kinds of end conditions. They also performed a finite element analysis of Timoshenko nano-beams based on the integral model of nonlocal continuum theory [13]. Ansari et al. [14] studied the free vibration and instability characteristics of nanoshells made of FGMs with internal fluid flow in thermal environment based on SGT. Yin et al. [15] investigated the vibration analysis of micro-pipes conveying fluid based on the MSGT. Hosseini and Bahaadini [16] inspected the stability behavior of sizedependent micro-pipes conveying fluid flow based on the MSGT. Ansari et al. [17] Studied the nonlinear vibration and instability of fluid-conveying SWBNNTs based on the MSGT. Based on the MCST, Dehrouyeh-Semnani et al. [18] performed the nonlinear dynamics of pipes containing fluid with non-uniform flow profile. Ansari et al. [19] studied the vibration and dynamic instability of cylindrical microshells made of functionally graded materials (FGMs) and containing flowing fluid based on MCST. Vibration and stability analysis of thin-walled fluidconveying pipe made of composite material was investigated by Bahaadini et al. [20]. Amabili and Garziera [21, 22] investigated the vibration of circular cylindrical shells with nonuniform edge constrains. Besides, there are many studies on the vibration and stability of micro/nano tubes conveying fluid [8, 23-31]. Recently, the nonlocal strain gradient theory which mixed the Aifantis’s strain gradient and Eringen’s nonlocal elasticity theories into a constitutive equation was introduced by Lim et al. [32]. They studied wave propagation in nano/micro beams model based on NSGT. Using NSGT, buckling [33], and vibration [34-36] of micro/nano structures have been studied. The linear and nonlinear vibrations of CNT in the context of NSGT were investigated in [37, 38].
4
Nonlinear vibration of nanobeam made of functionally graded materials by using NSGT was studied by Simsek [39]. Li and Hu [40] performed wave propagation in viscoelastic CNTs conveying fluid based on NSGT. By using nonlocal strain gradient Timoshenko and EulerBernoulli beams, Li et al. [41] inspected critical divergence velocity of micro-pipes conveying fluid. The influences of fluid mass ratio, surface effects, nonlocal and strain gradient parameters on the vibration of fluid- conveying nanotubes based on NSGT were examined by Atashafrooz et al. [42]. Choi et al. [43] performed the nonlinear vibration behavior of the thin-walled nano pipes conveying fluid. Vibration and instability of multi-walled CNTs conveying fluid were studied by Yun et al. [44] based on the NSGT. Dynamic stability and vibration of nanotubes conveying nanoflow based on NSGT were investigated by Bahaadini et al. [45, 46]. Besides, Bahaadini and Saidi [47] performed the stability of spinning thin-walled pipe reinforced with CNTs conveying fluid by using Galerkin method (GM). Dynamic instability analysis of the rotating thin-walled pipe reinforced with CNTs conveying fluid was analyzed by Bahaadini et al. [48]. Norouzzade et al. [49] studied the vibrational behavior of Timoshenko micro- and nano-beams By taking the nonlocal and strain gradient effects into account based on a novel size-dependent model. Norouzzade et al. [50] studied the static bending of nanoscale beams in the nonlinear regime based on NSGT. Norouzzade et al. [51] investigate the geometrically nonlinear wave propagation of nanobeams on the basis of the most comprehensive size dependent elasticity theory. Li et al. [52] investigated the size-dependent buckling analysis of nanobeams with a nonlocal strain gradient beam model incorporating the thickness effect. Tang et al. [53] investigate the static bending behavior of micro/nano-scale porous beams using a unified nonlocal strain gradient beam model with the thickness effect. Tang et al. [54] derived an
5
analytical solution for the bending frequencies of hinged-hinged nanobeams based on NSGT with the thickness effect. To the best of authors’ knowledge, no study investigated the vibration and flutter instability analysis of thin-walled nanotubes conveying magnetic nanoflow under magnetic field by using thin-walled Timoshenko beam theory. According to this theory, both flapping and lagging bending vibrations are considered and the proper components of the 3-D displacement vector are obtained. To formulate the size-dependent nanotubes conveying magnetic nanoflow, the nonlocal strain gradient thin-walled Timoshenko beam theory, magnetic nanoflow, Knudsen number and magnetic field are considered. The governing equations of motion are derived based on the extended Hamilton’s principle (EHP). Using the extended Galerkin method (EGM), the eigenvalue and flutter velocity are evaluated. The effects of strain gradient length scale, nonlocal parameter, Knudsen number and magnetic field on the natural frequencies and critical flutter velocity of the system are studied.
2. Theoretical formulation Based on NSGT, the total stress tensor is written as [32] ( )
where
(1)
denotes the one-dimensional differential operator, the classical stress tensor σ and the
higher-order stress tensor σ(1) are written as ( (
( (
) )
) )
(2)
( )
6
In equations (2) ε denote the classical strain tensor, ε represent the strain gradient tensor, C is the fourth-order elasticity tensor, constant and
is the material length scale parameter,
is an internal property of length. So,
is a material
displays the nonlocal parameter that
incorporates the small scale effects into the constitutive equations. At nanoscales, the material properties is size-dependent and the small scale effects of nanostructure have a crucial role in the dynamics of nanosystems. Therefore, it is necessary to use nonlocal theories incorporating small scale effects. Thus, the size-dependent constitutive equation for beams may be simplified as (
(
)
)
(3)
Fig. 1 is showed the model of the conveying-fluid thin-walled nanotube subjected to longitudinal magnetic field. the thin-walled nanotube has the length L, thickness h, outer diameter D, inner diameter d, and mass per unit length incompressible fluid of mass per unit length
. The nanotube conveying
, flowing axially with velocity
. In this study,
the nonlocal strain gradient thin-walled Timoshenko beam model is used to study vibration and stability analysis of the nanotube conveying nanoflow. Fig. 2 represents the kinematic variables and coordinate system of a thin-walled beam structure. The displacement components of thinwalled Timoshenko beams that undergo vertical bending with ignoring extension and torsion, can be written as follows [55, 56] (
)
(
),
(
)
(
),
(
)
(
)[ ( )
(4) ]
(
)[ ( )
7
]
In Eq. (4), the rigid body translations along the
- and
- axes are denoted by
(
), respectively. The rigid body rotations about the
(
) and
(
), respectively. The expressions for
- and (
) and
(
) and
- axes are represented by (
) can be written as
[56, 57] (
)
(
)
(
), (5)
(
)
(
where
and
)
(
)
are the shear strains and the partial derivative with respect to
is denoted by
superscript ( ). The non-zero strains components can be expressed as follows ,
,
,
The relations between strains of (
) and (
(6)
) in coordinate systems can be written as
,
(7)
,
Furthermore, the fluid particle velocity can be formulated as ([47] . ̇
/
. ̇
/
(
̇
̇
) ,
(8)
In Eq. (8), the subscript f shows the fluid and the superposed dot denotes the time derivative. The constitutive relations of thin-walled nanotubes based on NSGT can be formulated as
8
(
(
)
)
( [
)
]
(9) ][
[
]
where the reduced elastic coefficients can be written as ,
,
, (10)
, In Eq. (10), the Young and shear moduli and Poisson ratio of the nanotube are denoted by , ⁄
and , respectively. Furthermore,
denotes the shear correction coefficient.
The governing equations of motion and the corresponding boundary conditions are derived based on EHP which can be formulated as [58] ( ̇
∫ (
)
)
, (11)
at
,
In Eq. (11), the variation of the pipe kinetic energy, fluid kinetic energy and strain energy are denoted by
,
and
, respectively. Furthermore,
and
denote the position and
tangential vectors of a point on the free ended axis of the pipe, respectively. For thin-walled Timoshenko nanotubes, the virtual kinetic energy can be written as [55]
∫
∫
∫ {
̈
̈
[(
) ̈
(
) ̈ ]
[(
) ̈
(
) ̈ ]
9
(12) }
The virtual of kinetic energy of fluid flow is obtained as
∫
∫ { ,
∫
̈ ̇
̈
̈
,
̈ ̇
} (13)
∫ { , ̈
In Eq. (13),
and
̇ -
,
̈
̇ -
}
are the internal gyration radius about the x- and y- axes, respectively. The
virtual strain energy can be obtained as
)
∫ ∮ ∫ 0( ( )
.
( )
( )
( )
∫ ∮ ∫ 0.
( )
.
( )
where denoted by
⁄
/
/
/
1
/1
.
∫ ∮ 0.
( )
/
(14)
( )
( )
1
represents the one-dimensional differential operator. The classical stresses are ,
and
; the higher-order stresses are represented by
( )
,
( )
and
( )
.
By considering the components of strain and stress tensors obtained in Eqs. (9) and (14), the variation of strain is obtained as [55] 10
∫
{(
)
(
(
)
(
)
)
} (15)
( )
2. [
/
( )
. ( )
]
/
,
( )
( )
( )
3|
The components of Eq. (15) can be found in Appendix C. Considering the magnetic-fluid flow, the force applied to the nanotubes can be written as
( (
)
(16)
)
where the subscripts f and t correspond to the fluid and tube, respectively. In order to consider the small-size effect on fluid flow, a velocity correction factor ( (
by Rashidi et al. [9]. They used a modified nanotube by applying
)
(
where
)( (
is equal to 0.7 and
. In which
) was introduced as
/
(
)(
,
0.4 and
)(
)
)
as
(17)
is defined as: (
The magnetic force along the
)) (18)
. direction due to a longitudinal magnetic field based on the
Maxwell relation can be formulated as
11
4
where
5
(19)
represents the magnetic field permeability and
is the magnetic intensity of the
longitudinal magnetic field. By substituting Eqs. (12), (13), (15), (16) and (19) into Eq. (11), and using the arbitrary and independent properties of variations, the governing equations are derived as (
(
) (
)
) [(
) ̈
( ̇ (
δ
(
0
(
(
( )
(
(
)
(
) [(
) ̈
( ̇ (
(
)
(
)
)
) 1
( ]
)
(20)
(
) ̇ )
(
(
)
) (
)
) ̈
0
(
) ̇ )
( ( ̇
)
) 1
(
) ̇ )
(
)
(
) ]
(
)
(
)
)
)
(
(
) ̇ )
(
) ̈
( ̇
(
0(
(21) )
)
) (22)
(
)
(
) 1
(
) ̈
) [(
12
(
) ̈
̈ ]
) ̈
( (
) ̈
(
(
)
0(
̈
)
) (
(
)
) 1 (23)
(
) ̈
) [( ) ̈
(
) ̈
( ) ̈
(
̈ ] ̈
Besides, the boundary conditions are: at z=0:
(24)
at
( ) ( )
( )
( )
( )
( )
(25)
In equations of motion, the stiffness and reduced mass terms are represented by
and
,
respectively, which are defined in Appendix A.
3. Solution method The EGM is utilized to convert the partial differential equations into ordinary differential equations. Therefore, the following expansions are assumed: 13
̅ (
)
̅ (
,
)
, (26)
̅ (
)
̅ (
,
)
,
In Eqs. (26), the shape functions which satisfy the boundary conditions are represented by and
. The generalized coordinates vectors are denoted by
,
,
and
, , . The
following equation can be obtained by using EGM , - ̈( )
, - ̇( )
, - ( )
(27)
,
In Eq. (27), mass, damping and stiffness matrices are denoted as , - ,, - and , -, respectively. These matrices are defined in Appendix B. Defining the state vector as
( )
* ( ) ̇ ( )+ , Eq. (27) can be converted to the first-order
state space form as: ̇( )
, - ( ),
(28)
where , -
[
, , - , -
, ], , - , -
Substituting the state vector
( )
(29) ̅
(
)
into Eq. (28), the following equation can be
expressed: (, -
, -) ̅
The eigenvalue
(30)
. ( )
is a complex quantity, i.e.,
( ). The types of instability are
determined based on the sign of real and imaginary parts of the complex eigenvalue [27, 59].
14
4. Results and discussion The natural frequency and flutter velocity of conveying-fluid nanotubes subjected to longitudinal magnetic field based on the nonlocal strain gradient thin-walled Timoshenko theory are studied. The effects of fluid velocity, Knudsen number, strain gradient length scale, nonlocal parameter and magnetic field on the natural frequency and flutter velocity of the system are investigated. The dimensionless frequency as a function of fluid velocity for cantilever thinwalled nanotube conveying fluid is illustrated in Fig. 2 and the results are compared with those reported by Yun et al. [44] based on the classical thin-walled Timoshenko CNT. Fig. 2, shows that the present results are in excellent agreement with the reported data. In order to illustrate the accuracy of present study, the natural frequency of nanotube conveying fluid is compared those obtained using classical continuum theory. Also, for validation purposes and to demonstrate the accuracy of suggested model, the eigenfrequencies obtained from the current study are compared with those reported by Apuzzo et al. [60] in Table 4. Furthermore, the convergence of eigenvalues of thin-walled nanotube conveying fluid is shown in Table 1 for 𝛽 μ results at
, 𝜉
and and
,
,
. It should be noted that the small difference between the show that the eigenvalue with
is suitably accurate to study
the vibration and stability analysis of cantilever nanotube conveying nanflow. The effects of and Kn on the critical flutter velocity for different classical and non-classical theories is observed in Tables 2 and 3, respectively. It can be found that the critical flutter velocity increases by increasing the magnetic intensity
. While by increasing the Knudsen number, the critical
flutter velocity is decreased. Moreover, the maximum and minimum flutter velocities are predicted by using the SGT and NT Timoshenko thin-walled pipe models, respectively.
15
It should be noted that the non-classical continuum theories predict a more precise mechanical model on the nano-scale materials. Hence, the effects of non-classical continuum theories on the eigenvalues and critical flutter velocity are illustrated in Figs. 3 through 5. Furthermore, NSGT can be converted to a strain gradient theory (SGT) if 𝜉 nonlocal theory (NT) if 𝜉
and
and a classical theory (CT) if 𝜉
and and
,a .
Besides, it is seen that the natural frequencies and critical flutter velocity of the conveying-fluid thin-walled nanotubes based on the NSGT can be smaller than that of classical model, which can be described by the stiffness-softening effect owing to the values of scaling parameters. The stiffness-softening effects are seen when 𝜉
. In Fig. 3, the natural frequencies and flutter
velocity decrease by increasing the nonlocal parameter which means that considering the nonlocal effect leads to decrease the bending rigidity of the nanotubes. Besides in Fig. 4, the natural frequencies and flutter velocity increase by increasing the strain gradient parameter. Fig. 5 shows the effects of both strain gradient length scale and nonlocal parameters on the natural frequency and flutter velocity of the nanotubes containing magnetic nanoflow for 𝛽 and
,
. The results show that the natural frequency and flutter velocity decrease
as the nonlocal parameter increases which means that considering the nonlocal effect leads to decreasing the bending rigidity of the carbon nanotubes. Furthermore, the natural frequency and flutter velocity increases by increasing the strain gradient length scale. It means that the stiffness of the system increases, as the strain gradient length scale increases. Furthermore, the strain gradient and the nonlocal elasticity models predict stiffness enhancement effect and stiffness softening effect for the nanotube, respectively. The results show that the vibration and stability analyses of the thin-walled nanotubes based on the nonlocal strain gradient model may be more reliable.
16
The effect of aspect ratio ( ⁄ ) on the eigenvalues of thin-walled nanotubes conveying magnetic nanoflow under longitudinal magnetic field is investigated in Fig. 6. This figure is presented for 𝛽
𝜉
and
. The results show that the aspect
ratio effect tends to increase the bending stiffness of the thin-walled nanotubes and the natural frequencies and flutter velocity increase. The effects of intensity of longitudinal magnetic field on the eigenvalues of thin-walled nanotubes conveying magnetic nanoflow are studied in Fig. 7. The results are presented for 𝛽
,
and
𝜉
. The results indicated that the
natural frequencies and flutter velocity increase by increasing the intensity of longitudinal magnetic field. In other words, the results show that the magnetic fluid tends to increase the bending stiffness of the system and the natural frequency and flutter velocity increase. Figs. 8 and 9 shows the influence of Knudsen number on the natural frequencies and flutter velocity of magnetic nanoflow conveying-nanotube based on the NSGT for and 𝜉
. It is noted that the Helium gas is considered in Fig. 9. The Knudsen
number is used to identify different flow regimes such as the continuum flow regime ( ), the slip flow regime ( and the free molecular flow regime (
), the transition flow regime (
),
). For the continuum flow regime, it is revealed that
the natural frequencies and flutter velocity are not changed as the Knudsen number increases. Also, the natural frequencies and flutter velocity decrease, as the Knudsen number increases in the slip flow regime. Besides, the natural frequencies and flutter velocity decrease as the Knudsen number increases in the transient flow regime. For
, the natural frequencies
and flutter velocity tends to vanish and the effect of flow velocity on the vibration and stability analysis becomes faint. Fig. 10 shows the effects of mass fluid ratio on the eigenvalues of the
17
conveying-fluid nanotubes for
,
and
𝜉
. The results illustrate that the
natural frequency and flutter velocity increase as the mass fluid ratio increases. Fig. 11 shows the critical flutter velocity in terms of size-dependent parameters for different non-classical thin-walled beam models. In this figure, the values
.001, 𝛽
are considered. The results can be shown that the thin-walled pipe with SGT and NT models have the highest and lowest critical flutter velocity among others. Also, the results for thin-walled pipe with NSGT are tend to get closer to those of SGT or NT models due to size-dependent parameters. In this Figure, the flutter boundaries separate the stable and unstable areas. It should be noted that, in unstable area any initial dynamic structural disturbance will grow up till the mechanical failure will happen. Fig. 12 shows the effects of both strain gradient and nonlocal parameters on the natural frequency of the CNTs conveying nanoflow for
,𝛽
. It is clear that
the natural frequency decreases by increasing nonlocal parameter which means that the considering nonlocal elasticity theory leads to decrease the bending rigidity of the CNTs. Whereas, by increasing the strain gradient parameter, the stiffness of the CNTs increases, and consequently, the natural frequency increases.
5. Conclusion The size-dependent stability analysis of the conveying-fluid thin-walled nanotubes under longitudinal magnetic field was investigated. Using NSGT, thin-walled Timoshenko beam theory, Knudsen number and magnetic nanoflow, the equations of motion were derived based on the EHP. Using the EGM, these equations were discretized. The influences of strain gradient 18
length scale, nonlocal parameter, fluid velocity, Knudsen number and magnetic field on the vibration and stability analysis of the system were discussed. It was observed that the magnetic fluid has considerable effect on the natural frequency and flutter velocity of the nanotubes. Specifically, this study showed that the magnetic field effects increase the critical flow velocity, thus makes the system more stable. It was demonstrated that as the Knudsen number rises, the natural frequency and critical flutter velocity decreases. Besides, it was shown that the natural frequency and critical flow velocity decrease with increasing nonlocal parameter, whereas those have the opposite effect with increasing strain gradient parameter. Furthermore, it was seen that in the NSGT, if the nonlocal coefficient be greater than the strain gradient parameter, by increasing the nonlocal strain gradient parameter, the natural frequency and consequently the critical flow velocity decreases; otherwise, it increases. It was illustrated that increasing the fluid mass ratio and aspect ratio of the CNT increase the natural frequency of the system, which makes the system more stable.
Appendix A The global stiffness quantities
counting the pretwist effect can be expressed as bellows
∮
∮(
∮4
)
(
∮(
) 5
)
∮4
19
(
)
(A.1)
(
) 5
∮4
(
) 5
∮4
(
)
(
) 5
where
(A.2) {
}
*
∫
+
Furthermore, the mass terms (
)
∮
(
(
)
∮
4.
(
)
∫
(
are presented as )
/ . / . / 5
(A.3)
,
)
Appendix B To simplify the analysis, we introduce the following dimensionless parameters as below ̅
̅ ̅
̅ ̅
̅
𝛽
√
(
(B.1)
√
)
𝛽
(
20
)
(
)
√
(
(
)
)
√ 𝜉
The matrices are defined as follows
, -
, [
]
, [
] (B.2)
, [
(𝛽
𝛽)
∫ * + * +
]
(
) (𝛽
𝛽)
∫ * + * +
(B.3) √𝛽 (
)
∫ * + * +
(
) 𝛽
∫ * +
* +
∫ * + * +
21
∫ * + * +
∫ (* + * + (
) [
* +
∫ .{
(𝛽
√𝛽 (
* + * +
* +
(
} * +
{
∫ * + * +
* + * +
* + * + ]
* + * + )
} * + /
(
(
) 𝛽
) (𝛽
𝛽)
∫ * +
* +
∫ * + * +
∫ * + * +
∫ * + * +
)
∫ (* + * + (
) * + * +
) * + * +
∫ * + * +
𝛽)
(
) [
* +
∫ (*
(
* + * +
* +
+ * +
) ∫ .{
(
*
(
) * + * +
) * + * +
* + * +
* + * + ]
* + * + )
+ * + )
} {
}
(
)
(
22
){
} {
}/
+ {
∫ .*
∫ .
{
} {
}
∫ .* + {
) ∫ (*
+ *
}
* +
} {
}/
0{
{
} {
}
{
}
{
} 1/
+
*
+ 1/
}/
+
}
{
+ *
} *
∫ .{
*
} {
+ {
}
{
(
∫ .
{
) *
(
}
∫ * + {
∫ (
}
+
*
+
(
(
+ *
) {
+
)
}
{
(
} )
)*
} *
+/
0*
+ *
23
+
+ *
+)
*
∫ (* + *
+
* +
∫ * + *
*
∫ (
*
+)
+
+ *
+
*
+
*
+ )
Appendix C The stress resultants are defined as follows
(
)
(
∫
)
.
( )
( )
/
∫
( )
(
) (C.1)
(
)
Where
(
∫
)
( )
.
( )
/
.
∫
( )
denotes the axial normal stress and ( )
( )
/
denotes the shear stress
,
( )
(C.2)
( )
Moreover, the other stress resultants and the stress-couples can be defined as (
)
( )
∮
(
)
∮
( )
(C.3) (
)
∮0
( )
( )
1
( )
(
)
24
∮0
( )
( )
( )
( )
1
(
)
∮0
(
)
∮6
(
)
∮6
where
( )
( )
( )
( )
1
7
( )
7
)
∮0
( )
( )
( )
( )
( )
(
)
∮6
( )
( )
( )
( )
(
)
∮6
( )
( )
( )
corresponds to the axial force;
and y- directions, respectively;
(
and
and
1
7 7
denote associated with the shear forces in xrepresent the bending moments of x- and y-
directions, respectively.
25
References [1] S. Iijima, Helical microtubules of graphitic carbon, nature, 354 (1991) 56. [2] G. Hummer, J.C. Rasaiah, J.P. Noworyta, Water conduction through the hydrophobic channel of a carbon nanotube, Nature, 414 (2001) 188. [3] Y. Gao, Y. Bando, Nanotechnology: carbon nanothermometer containing gallium, Nature, 415 (2002) 599. [4] R.F. Gibson, E.O. Ayorinde, Y.F. Wen, Vibrations of carbon nanotubes and their composites: A review, Composites Science and Technology, 67 (2007) 1-28. [5] D. Mattia, Y. Gogotsi, Review: static and dynamic behavior of liquids inside carbon nanotubes, Microfluidics and Nanofluidics, 5 (2008) 289-305. [6] D. Pantarotto, C.D. Partidos, J. Hoebeke, F. Brown, E. Kramer, J.P. Briand, S. Muller, M. Prato, A. Bianco, Immunization with peptide-functionalized carbon nanotubes enhances virusspecific neutralizing antibody responses, Chemistry & biology, 10 (2003) 961-966. [7] E. Ghavanloo, S.A. Fazelzadeh, Flow-thermoelastic vibration and instability analysis of viscoelastic carbon nanotubes embedded in viscous fluid, Physica E: Low-dimensional Systems and Nanostructures, 44 (2011) 17-24. [8] R. Bahaadini, M. Hosseini, Nonlocal divergence and flutter instability analysis of embedded fluid-conveying carbon nanotube under magnetic field, Microfluidics and Nanofluidics, 20 (2016) 108. [9] V. Rashidi, H.R. Mirdamadi, E. Shirani, A novel model for vibrations of nanotubes conveying nanoflow, Computational Materials Science, 51 (2012) 347-352. [10] R. Bahaadini, M. Hosseini, Effects of nonlocal elasticity and slip condition on vibration and stability analysis of viscoelastic cantilever carbon nanotubes conveying fluid, Computational Materials Science, 114 (2016) 151-159. [11] N. Challamel, Z. Zhang, C.M. Wang, J.N. Reddy, Q. Wang, T. Michelitsch, B. Collet, On nonconservativeness of Eringen’s nonlocal elasticity in beam mechanics: correction from a discrete-based approach, Archive of Applied Mechanics, 84 (2014) 1275-1292. [12] A. Norouzzadeh, R. Ansari, H. Rouhi, Pre-buckling responses of Timoshenko nanobeams based on the integral and differential models of nonlocal elasticity: An isogeometric approach, Applied Physics A, 123 (2017) 330. [13] A. Norouzzadeh, R. Ansari, Finite element analysis of nano-scale Timoshenko beams using the integral model of nonlocal elasticity, Physica E: Low-dimensional Systems and Nanostructures, 88 (2017) 194-200. [14] R. Ansari, R. Gholami, A. Norouzzadeh, Size-dependent thermo-mechanical vibration and instability of conveying fluid functionally graded nanoshells based on Mindlin's strain gradient theory, Thin-Walled Structures, 105 (2016) 172-184. [15] L. Yin, Q. Qian, L. Wang, Strain gradient beam model for dynamics of microscale pipes conveying fluid, Applied Mathematical Modelling, 35 (2011) 2864-2873. [16] M. Hosseini, R. Bahaadini, Size dependent stability analysis of cantilever micro-pipes conveying fluid based on modified strain gradient theory, International Journal of Engineering Science, 101 (2016) 1-13. [17] R. Ansari, A. Norouzzadeh, R. Gholami, M.F. Shojaei, M. Hosseinzadeh, Size-dependent nonlinear vibration and instability of embedded fluid-conveying SWBNNTs in thermal environment, Physica E: Low-dimensional Systems and Nanostructures, 61 (2014) 148-157.
26
[18] A.M. Dehrouyeh-Semnani, H. Zafari-Koloukhi, E. Dehdashti, M. Nikkhah-Bahrami, A parametric study on nonlinear flow-induced dynamics of a fluid-conveying cantilevered pipe in post-flutter region from macro to micro scale, International Journal of Non-Linear Mechanics, 85 (2016) 207-225. [19] R. Ansari, R. Gholami, A. Norouzzadeh, S. Sahmani, Size-dependent vibration and instability of fluid-conveying functionally graded microshells based on the modified couple stress theory, Microfluidics and nanofluidics, 19 (2015) 509-522. [20] R. Bahaadini, M.R. Dashtbayazi, M. Hosseini, Z. Khalili-Parizi, Stability analysis of composite thin-walled pipes conveying fluid, Ocean Engineering, 160 (2018) 311-323. [21] M. Amabili, R. Garziera, Vibrations of circular cylindrical shells with nonuniform constraints, elastic bed and added mass; Part I: empty and fluid-filled shells, Journal of Fluids and Structures, 14 (2000) 669-690. [22] M. Amabili, R. Garziera, Vibrations of circular cylindrical shells with nonuniform constraints, elastic bed and added mass. Part II: shells containing or immersed in axial flow, Journal of Fluids and Structures, 16 (2002) 31-51. [23] M. Hosseini, R. Bahaadini, M. Makkiabadi, Application of the Green function method to flow-thermoelastic forced vibration analysis of viscoelastic carbon nanotubes, Microfluidics and Nanofluidics, 22 (2018) 6. [24] R. Bahaadini, M. Hosseini, Flow-induced and mechanical stability of cantilever carbon nanotubes subjected to an axial compressive load, Applied Mathematical Modelling, 59 (2018) 597-613. [25] R. Bahaadini, A.R. Saidi, M. Hosseini, Dynamic stability of fluid-conveying thin-walled rotating pipes reinforced with functionally graded carbon nanotubes, Acta Mechanica, 229 (2018) 5013-5029. [26] M. Hosseini, A.Z.B. Maryam, R. Bahaadini, Forced vibrations of fluid-conveyed double piezoelectric functionally graded micropipes subjected to moving load, Microfluidics and Nanofluidics, 21 (2017) 134. [27] R. Bahaadini, M. Hosseini, A. Jamalpoor, Nonlocal and surface effects on the flutter instability of cantilevered nanotubes conveying fluid subjected to follower forces, Physica B: Condensed Matter, 509 (2017) 55-61. [28] M. Mohammadimehr, M. Mehrabi, Electro-thermo-mechanical vibration and stability analyses of double-bonded micro composite sandwich piezoelectric tubes conveying fluid flow, Applied Mathematical Modelling, 60 (2018) 255-272. [29] B. Wang, Z. Deng, H. Ouyang, X. Xu, Free vibration of wavy single-walled fluid-conveying carbon nanotubes in multi-physics fields, Applied Mathematical Modelling, 39 (2015) 67806792. [30] M. Hosseini, M. Sadeghi-Goughari, Vibration and instability analysis of nanotubes conveying fluid subjected to a longitudinal magnetic field, Applied Mathematical Modelling, 40 (2016) 2560-2576. [31] A. Ghorbanpour Arani, R. Kolahchi, Z. Khoddami Maraghi, Nonlinear vibration and instability of embedded double-walled boron nitride nanotubes based on nonlocal cylindrical shell theory, Applied Mathematical Modelling, 37 (2013) 7685-7707. [32] C.W. Lim, G. Zhang, J.N. Reddy, A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation, Journal of the Mechanics and Physics of Solids, 78 (2015) 298-313.
27
[33] A. Farajpour, M.H. Yazdi, A. Rastgoo, M. Mohammadi, A higher-order nonlocal strain gradient plate model for buckling of orthotropic nanoplates in thermal environment, Acta Mechanica, 227 (2016) 1849-1867. [34] Y. Tang, Y. Liu, D. Zhao, Viscoelastic wave propagation in the viscoelastic single walled carbon nanotubes based on nonlocal strain gradient theory, Physica E: Low-dimensional Systems and Nanostructures, 84 (2016) 202-208. [35] H. Zeighampour, Y. Tadi Beni, I. Karimipour, Material length scale and nonlocal effects on the wave propagation of composite laminated cylindrical micro/nanoshells, The European Physical Journal Plus, 132 (2017) 503. [36] L. Lu, X. Guo, J. Zhao, Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory, International Journal of Engineering Science, 116 (2017) 12-24. [37] X. Li, Y. Li, Y. Qin, Free vibration characteristics of a spinning composite thin-walled beam under hygrothermal environment, International Journal of Mechanical Sciences, 119 (2016) 253-265. [38] R. Fernandes, S. El-Borgi, S. Mousavi, J.N. Reddy, A. Mechmoum, Nonlinear sizedependent longitudinal vibration of carbon nanotubes embedded in an elastic medium, Physica E: Low-dimensional Systems and Nanostructures, 88 (2017) 18-25. [39] M. Şimşek, Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach, International Journal of Engineering Science, 105 (2016) 12-27. [40] L. Li, Y. Hu, Wave propagation in fluid-conveying viscoelastic carbon nanotubes based on nonlocal strain gradient theory, Computational Materials Science, 112 (2016) 282-288. [41] M. Mohammadimehr, M.J. Farahi, S. Alimirzaei, Vibration and wave propagation analysis of twisted micro-beam using strain gradient theory, Applied Mathematics and Mechanics, 37 (2016) 1375-1392. [42] M. Atashafrooz, R. Bahaadini, H.R. Sheibani, Nonlocal, strain gradient and surface effects on vibration and instability of nanotubes conveying nanoflow, Mechanics of Advanced Materials and Structures, (2018) 1-13. [43] J. Choi, O. Song, S.K. Kim, Nonlinear stability characteristics of carbon nanotubes conveying fluids, Acta Mechanica, 224 (2013) 1383-1396. [44] K. Yun, J. Choi, S.K. Kim, O. Song, Flow-induced vibration and stability analysis of multiwall carbon nanotubes, Journal of mechanical science and technology, 26 (2012) 3911-3920. [45] R. Bahaadini, A.R. Saidi, M. Hosseini, On dynamics of nanotubes conveying nanoflow, International Journal of Engineering Science, 123 (2018) 181-196. [46] R. Bahaadini, A.R. Saidi, M. Hosseini, Flow-induced vibration and stability analysis of carbon nanotubes based on the nonlocal strain gradient Timoshenko beam theory, Journal of Vibration and Control, 21 (2018) 203-218. [47] R. Bahaadini, A.R. Saidi, Stability analysis of thin-walled spinning reinforced pipes conveying fluid in thermal environment, European Journal of Mechanics-A/Solids, 72 (2018) 298-309. [48] R. Bahaadini, A.R. Saidi, M. Hosseini, Dynamic stability of fluid-conveying thin-walled rotating pipes reinforced with functionally graded carbon nanotubes, Acta Mechanica, (2018) 117. [49] A. Norouzzadeh, R. Ansari, H. Rouhi, Isogeometric vibration analysis of small-scale Timoshenko beams based on the most comprehensive size-dependent theory, Scientia Iranica, 25 (2018) 1864-1878. 28
[50] A. Norouzzadeh, R. Ansari, H. Rouhi, Nonlinear bending analysis of nanobeams based on the nonlocal strain gradient model using an isogeometric finite element approach, Iranian Journal of Science and Technology, Transactions of Civil Engineering, 43 (2019) 533-547. [51] A. Norouzzadeh, R. Ansari, H. Rouhi, Nonlinear wave propagation analysis in Timoshenko nano-beams considering nonlocal and strain gradient effects, Meccanica, 53 (2018) 3415-3435. [52] L. Li, H. Tang, Y. Hu, The effect of thickness on the mechanics of nanobeams, International Journal of Engineering Science, 123 (2018) 81-91. [53] H. Tang, L. Li, Y. Hu, Coupling effect of thickness and shear deformation on sizedependent bending of micro/nano-scale porous beams, Applied Mathematical Modelling, 66 (2019) 527-547. [54] H. Tang, L. Li, Y. Hu, W. Meng, K. Duan, Vibration of nonlocal strain gradient beams incorporating Poisson's ratio and thickness effects, Thin-Walled Structures, 137 (2019) 377-391. [55] L. Librescu, O. Song, Thin-walled composite beams: theory and application, Springer Science & Business Media 2005. [56] R. Bahaadini, A.R. Saidi, On the stability of spinning thin-walled porous beams, ThinWalled Structures, 132 (2018) 604-615. [57] L. Librescu, S.Y. Oh, O. Song, Thin-walled beams made of functionally graded materials and operating in a high temperature environment: vibration and stability, Journal of Thermal Stresses, 28 (2005) 649-712. [58] M.P. Paidoussis, Fluid-structure interactions: slender structures and axial flow, Academic press1998. [59] R. Bahaadini, A.R. Saidi, Aeroelastic analysis of functionally graded rotating blades reinforced with graphene nanoplatelets in supersonic flow, Aerospace Science and Technology, 80 (2018) 381-391. [60] A. Apuzzo, R. Barretta, S. Faghidian, R. Luciano, F.M. de Sciarra, Free vibrations of elastic beams by modified nonlocal strain gradient theory, International Journal of Engineering Science, 133 (2018) 99-108.
29
Table Captions Table 1. The convergence of eigenvalues of nanotubes conveying nanflow (for β ,𝜉
,μ
and
,
).
Table 2. The critical flutter velocity of cantilever nanotubes conveying magnetic nanoflow (for 𝛽 , , 𝜉 ). Table 3. The critical flutter velocity of cantilever nanotubes conveying magnetic nanoflow (for 𝛽 , , 𝜉 ). Table 4. The comparison of the frequencies obtained from the present model with the results presented by Apuzzo et al. [60].
Figure Captions Fig. 1. Schematic of a nanotube conveying magnetic nanoflow under magnetic field. Fig. 2. The comparison between the results of present study and those reported by Yun et al. [44]. Fig. 3. The variation of (a) natural frequency (b) and real part of eigenvalues for 𝛽 𝜉
,
.
Fig. 4. The variation of (a) natural frequency (b) and real part of eigenvalues for 𝛽
Fig. 5. The Argand diagram of the thin-walled carbon nanotubes for 𝛽
,
and
. Fig. 6. The variation of (a) natural frequency (b) and real part of eigenvalues for 𝛽 ,𝜉
,
and
.
Fig. 7. The variation of (a) natural frequency (b) and real part of eigenvalues for 𝛽 and
𝜉
,
.
Fig. 8. The variation of (a) natural frequency (b) and real part of eigenvalues for 𝛽 . 30
,
Fig. 9. The variation of (a) natural frequency (b) and real part of eigenvalues for and 𝜉
,
.
Fig. 10. The Argand diagram of the thin-walled carbon nanotubes for 𝜉
,
,
and
.
Fig. 11. The critical flutter velocity in terms of size-dependent parameters for thin-walled carbon nanotubes ( .001, 𝛽 ). Fig. 12. Variation of natural frequency versus nanlocal and strain gradient parameters for thinwalled carbon nanotubes ( ,𝛽 ).
Table 1. The convergence of eigenvalues of nanotubes conveying nanflow (for 𝛽 ,𝜉
,
and
).
N 4
3.392695-i2.096368
20.607984-i1.850850
58.835530-i1.916890
5
3.384872-i2.099706
20.521909-i1.882895
58.381143-i1.792144
6
3.387659-i2.098219
20.485642-i1.871531
57.945738-i1.865083
7
3.386026-i2.099021
20.480879-i1.875274
57.896421-i1.849693
8
3.386670-i2.098572
20.472613-i1.873468
57.833426-i1.863386
9
3.386631-i2.098680
20.470710-i1.873991
57.821514-i1.863896
10
3.386359-i2.098791
20.471034-i1.873763
57.806043-i1.862834
31
,
Table 2. The critical flutter velocity of cantilever nanotubes conveying magnetic nanoflow (for 𝛽 , , 𝜉 ). ( ) CT
NT
SGT
NSGT
0
4.3346
4.3168
4.3627
4.4239
50
4.6824
4.6657
4.7084
4.7650
100
5.5972
5.5829
5.6119
5.6662
150
6.8553
6.8432
6.8730
6.9111
Table 3. The critical flutter velocity of cantilever nanotubes conveying magnetic nanoflow (for 𝛽 , , 𝜉 ).
CT
NT
SGT
NSGT
0
4.3346
4.3168
4.3627
4.4239
0.001
4.3042
4.2865
4.3321
4.3928
0.01
4.0708
4.0540
4.0972
4.1546
0.1
3.0280
3.0156
3.0476
3.0903
Table 4. The comparison of the frequencies obtained from the present model with the results presented by Apuzzo et al. [60]. Natural frequencies Present study μ 0 0. 2 0. 4 0. 6 0. 8 1
𝜉 1.0084595 3 0.7352661 6 0.5723993 4 0.4673513 5 0.3939144 4 0.3400715 8
𝜉 1.096907 0.834476 2 0.670989 8 0.560478 1 0.480777 0.420533
Apuzzo et al. [60]
𝜉
𝜉
1.164200 6 0.910181 8 0.751318 2 0.641014 3 0.559740 3 0.496819 3
1.201338 8 0.954117 9 0.800682 4 0.693235 1 0.613287 1 0.550564 3
𝜉 1.227948 2 0.987460 9 0.839662 2 0.736334 4 0.659219 7 0.598238 4
32
𝜉
𝜉
𝜉
𝜉
𝜉
1.01778
1.10709
1.17522
1.21277
1.24021
0.74257 8
0.84290 5 0.67822 2 0.56680 7 0.48660 5 0.42618 1
0.91996 6 0.75956 3 0.64858 3 0.56674 8 0.50369 8
0.96459 1 0.80955 9 0.70163 7 0.62122 3 0.55853 1
0.99841 9 0.84912 8 0.74540 2 0.66789 4 0.60714 2
0.57857 0.47234 1 0.39849 3 0.34436 4
Fig. 1. Schematic of a nanotube conveying magnetic nanoflow under magnetic field.
Fig. 2. The comparison between the results of present study and those reported by Yun et al.
33
[44].
(a)
(b) Fig. 3. The variation of (a) natural frequency (b) and real part of eigenvalues for 𝛽 34
,
𝜉
.
(a)
(b) Fig. 4. The variation of (a) natural frequency (b) and real part of eigenvalues for 𝛽 35
,
.
Fig. 5. The Argand diagram of the thin-walled carbon nanotubes for 𝛽 .
36
and
(a)
(b) Fig. 6. The variation of (a) natural frequency (b) and real part of eigenvalues for 𝛽 𝜉
and
.
37
(a)
(b) Fig. 7. The variation of (a) natural frequency (b) and real part of eigenvalues for 𝛽 𝜉
38
,
(a)
(b) Fig. 8. The variation of (a) natural frequency (b) and real part of eigenvalues for 𝛽 .
39
(a)
(b) Fig. 9. The variation of (a) natural frequency (b) and real part of eigenvalues for .
40
,
Fig. 10. The Argand diagram of the thin-walled carbon nanotubes for 𝜉
.
41
and
7 = 0, (NT) = 0, (SGT) = 0.1, (NSGT) = 0.1, (NSGT)
6.8 6.6
Flutter
Critical flutter velocity, (ucr )
6.4 6.2 6
5.8 5.6 5.4 5.2 5
Stable
4.8 4.6 0
0.05
0.1
0.15
0.2
Size-dependent parameters, (,)
Fig. 11. The critical flutter velocity in terms of size-dependent parameters for thin-walled carbon nanotubes ( ,𝛽 ).
42
Fig. 12. Variation of natural frequency versus nanlocal and strain gradient parameters for thinwalled carbon nanotubes ( ,𝛽 ).
43